The models in this study are primarily based upon the models introduced in Chen et al. (1986) and Fama and French (1993, 2015) and requires the use of both macroeconomic and characteristic-based variables. After completing the necessary transformations of variables from levels to changes and changes in returns using logarithmic differences and demeaning of variables, we end up with a total of 329 monthly observations spanning from August 1990 to December 2017. We provide a comprehensive description of the transformations of the variables in the subsequent subsection (see also Table 1 for a summary), whereas the procedure of demeaning the variables is previously described in the
methodology section. Moreover, we have adopted the use of logarithmic returns in the macroeconomic variables following Chen et al. (1986).
We have not found a sufficient amount of financial data for Norwegian firms prior to 1995 and have therefore created the factors RMW and CMA for the Norwegian stock market only from January 1995 to December 2017. This results in a total of 276 observations for the CMA and RMW variables.
5.1 Test Assets
We have collected the monthly returns on twenty-eight value-weighted portfolios that are sorted by B/M, momentum and industry to apply as main test assets in the main models. Ten portfolios are sorted by book-to-market, ten portfolios are sorted by momentum, whereas eight portfolios are sorted by industry, due to lack of sufficient data for two of the industry portfolios. The test assets are collected in
full through Ødegaard4, and the portfolios are sorted by similar criteria used to generate the factor portfolios. Estimates of the monthly the risk-free rate 𝑅𝐹𝑡 in the whole sample period are also collected through Ødegaard, and is subtracted from the monthly test asset returns 𝑟𝑖,𝑡 to obtain the monthly excess returns of the portfolios 𝑅𝑖,𝑡:
𝑅𝑖,𝑡 = 𝑟𝑖,𝑡− 𝑅𝐹𝑡. (24)
We have also obtained the monthly returns on the portfolios used as test assets (BS, ISM, ISB and SBM) in the robustness analysis through Ødegaard.
5.2 Risk Factors
The basic time-series data required to create the macroeconomic factors are collected from various sources. The variables are collected and calculated without considering the growth in inflation, i.e. they are in nominal terms. In the following is a description of how each of the variables are constructed, and a summary of the definitions, sources and transformations of the factors are found in Table 1.
Table 1: Definitions of Series and Transformations
Variables Definitions of basic series and sources
CPI Natural logarithm of the Consumer Price Index (Statistics Norway).*
OILPRICE Natural logarithm of the futures price of North Sea Brent Crude oil (LCOc1) (Macrobond).
MARKET Natural logarithm of the closing price for the Oslo Børs All-share index (Macrobond).
USD/NOK Natural logarithm of the USD/NOK exchange rate (Norges Bank).
10Y Natural logarithm of the monthly average of daily quotes on Norwegian 10-year government bonds (Norges Bank).
3Y Natural logarithm of the monthly average of daily quotes on Norwegian 3-year government bonds (Norges Bank).
INDPROD Natural logarithm of the index of production, manufacturing ex. petroleum-related industries (Statistics Norway).*
CGI Natural logarithm of the domestic trade, households consumption of goods index (Macrobond).*
4 Bernt Arne Ødegaard have provided public asset pricing data for the Oslo Stock Exchange.
Retrieved from: http://finance.bi.no/~bernt/financial_data/index.html
HML Factor portfolio as calculated by Fama and French using Norwegian data (Ødegaard).
SMB Factor portfolio as calculated by Fama and French using Norwegian data (Ødegaard).
RMW Factor portfolio as calculated by Fama and French using Norwegian data.
Construction of the factor is described in detail in section 5.3.
CMA Factor portfolio as calculated by Fama and French using Norwegian data.
Construction of the factor is described in detail in section 5.3.
*Seasonally adjusted series.
Transformations Definitions of transformations
It = CPIt – CPIt-1 Monthly inflation.
INFt = It – It-1 Monthly change in inflation.
OILt = OILPRICEt – OILPRICEt-1 Monthly change in the price of North Sea Brent Crude oil.
MKTt = MARKETt – MARKETt-1 Monthly return on the Oslo Børs All-share index.
EMKTt = MKTt – RFt Excess monthly return on the Oslo Børs All-share index.
FXt = USD/NOKt – USD/NOKt-1 Monthly change in the USD/NOK exchange rate.
3GBt = 3Yt – 3Yt-1 Monthly change in 3-year government bonds.
10GBt = 10Yt – 10Yt-1 Monthly change in 10-year government bonds.
TSt = 10GBt – 3GBt-1 Monthly term spread between the 10-year and 3-year bonds.
CONt = CGIt – CGIt-1 Monthly change in consumption.
IPt = INDPRODt – INDPRODt-1 Monthly growth rate of Norwegian industrial production.
5.2.1 Inflation
The consumer price index for Norway is collected through Statistics Norway and yields a monthly seasonally adjusted time series. After taking the natural
logarithm of the prices, we obtain the series 𝐶𝑃𝐼. The monthly inflation, 𝐼𝑡 is then computed as:
𝐼𝑡 = 𝐶𝑃𝐼𝑡− 𝐶𝑃𝐼𝑡−1 (25)
where the subscript 𝑡 denotes the CPI value at the end of time 𝑡, whereas 𝑡 − 1 denotes the one-month antecedent CPI value. This subscript convention is adopted throughout the study.
Further, by taking the first difference of the inflation series:
𝐼𝑁𝐹𝑡 = 𝐼𝑡− 𝐼𝑡−1 (26) we obtain the series of unexpected monthly changes in inflation, as we assume that the expected value of 𝐼𝑁𝐹𝑡 at time 𝑡 − 1 is equal to 𝐼𝑁𝐹𝑡−1 i.e.
𝐸[𝐼𝑁𝐹𝑡|𝐼𝑁𝐹𝑡−1] − 𝐼𝑁𝐹𝑡= 0. This expectation is also assumed for the following factors.
5.2.2 Oil price
The price series of LCOc1 oil futures contracts are collected through Macrobond and yields monthly closing prices for ICE Brent Crude oil denominated in US dollars (USD). We obtain the series 𝑂𝐼𝐿𝑃𝑅𝐼𝐶𝐸 by taking the natural logarithm of the monthly prices. Furthermore, the monthly change in the price of crude oil is computed as:
𝑂𝐼𝐿𝑡 = 𝑂𝐼𝐿𝑃𝑅𝐼𝐶𝐸𝑡− 𝑂𝐼𝐿𝑃𝑅𝐼𝐶𝐸𝑡−1 . (27)
We follow the suggestion of Boyer and Filion (2007) that preserving this
denomination will enable us to identify and isolate the impact of variations in the exchange rate independently of variations in the oil prices. Further, Brent Crude oil contracts are used rather than WTI, as European oil production tends to be priced relative to this oil (Bjørnland, 2009).
5.2.3 Market index
The series of monthly closing prices on the Oslo All-share index (OSEAX) are collected through Macrobond to proxy for the market return in Norway. The OSEAX is a value-weighted index that comprise all shares listed on Oslo Stock Exchange and it is adjusted for dividend payments (Oslo Børs, 2018). Firstly, we take the natural logarithm of the prices and obtain the series 𝑀𝐴𝑅𝐾𝐸𝑇. Secondly, the monthly return on the market index is calculated as the first difference of the 𝑀𝐴𝑅𝐾𝐸𝑇 series:
𝑀𝐾𝑇𝑡= 𝑀𝐴𝑅𝐾𝐸𝑇𝑡− 𝑀𝐴𝑅𝐾𝐸𝑇𝑡−1. (28)
Lastly, the monthly excess return on the market index is calculated by subtracting the risk-free rate:
𝐸𝑀𝐾𝑇𝑡= 𝑀𝐾𝑇𝑡− 𝑅𝐹𝑡 (29)
where 𝑅𝐹𝑡 is the risk-free rate collected from Ødegaard.
5.2.4 Exchange rate
The monthly series of the exchange rate between the Norwegian krone and US dollar are collected through Norges Bank. The exchange rates are calculated by Norges Bank as monthly averages of the mid-points between bid and ask rates in the interbank market at a given time (Norges Bank, 2018). By taking the natural logarithm of the exchange rate series we obtain the series 𝑈𝑆𝐷/𝑁𝑂𝐾. The changes in the
𝑈𝑆𝐷/𝑁𝑂𝐾 exchange rates are then calculated as:
𝐹𝑋𝑡 = 𝑈𝑆𝐷𝑡/𝑁𝑂𝐾𝑡− 𝑈𝑆𝐷𝑡−1/𝑁𝑂𝐾𝑡−1 . (30)
5.2.5 Term spread
We have collected the monthly average of daily quotes on government bonds from Norges Bank for bonds with a maturity of ten and three years. By taking the natural logarithm of the 3-year and 10-year bond quotes, we obtain the series 3𝑌 and 10𝑌, respectively. The logarithmic returns for the series are then generated as the first differences:
3𝐺𝐵𝑡 = 3𝑌𝑡− 3𝑌𝑡−1 (31)
10𝐺𝐵𝑡 = 10𝑌𝑡− 10𝑌𝑡−1 . (32)
The term spread variable is then calculated as:
𝑇𝑆𝑡 = 10𝐺𝐵𝑡− 3𝐺𝐵𝑡−1 . (33)
We nevertheless note that although we calculate the term spread in similarity to Chen et al. (1986), we do not apply the yield on a treasury bill, but rather a 3-year government bond as the subtrahend. This is due to lack of treasury bills data from Norges Bank prior to February 2003.
5.2.6 Consumption
The households’ consumption of goods index for the domestic trade in Norway is collected through Macrobond and yields a monthly seasonally adjusted time series. The consumption series 𝐶𝐺𝐼 are constructed by taking the natural logarithm of the time series. Taking the first difference of the 𝐶𝐺𝐼 series:
𝐶𝑂𝑁𝑡 = 𝐶𝐺𝐼𝑡− 𝐶𝐺𝐼𝑡−1 (34)
yields the growth rates in nominal household consumption. However, the consumption figures are not disclosed until one month after the time of the observation. Therefore, to make the variable contemporaneous with the other variables we adopt the approach of Gjerde and Saettem (1999) to led the variable one period.
5.2.7 Industrial production
The index of production for Norway is collected through Statistics Norway and yields a monthly seasonally adjusted time series. The collected production index focuses primarily on the manufacturing industry and excludes petroleum-related industries. Taking the natural logarithm of the collected index of production, we obtain the series 𝐼𝑁𝐷𝑃𝑅𝑂𝐷. The industrial production factor is then constructed as: 𝐼𝑃𝑡= 𝐼𝑁𝐷𝑃𝑅𝑂𝐷𝑡 − 𝐼𝑁𝐷𝑃𝑅𝑂𝐷𝑡−1 (35)
which yields the monthly growth rate in the Norwegian industrial production. In similarity to the consumption variable, we follow Chen et al. (1986) and Gjerde and Saettem (1999) and allow the subsequent statistical work to lead it by one month.
5.2.8 HML and SMB
We have collected monthly returns for the Fama and French benchmark factors 𝐻𝑀𝐿 and 𝑆𝑀𝐵 through Ødegaard, which are calculated for the Norwegian stock market. Ødegaard constructs the 𝐻𝑀𝐿 and 𝑆𝑀𝐵 factors by double sorting the stocks on the Norwegian stock market into six portfolios and further compute the factors as:
𝑆𝑀𝐵 = 𝑆𝑉+𝑆𝑁+𝑆𝐺3 −𝐵𝑉+𝐵𝑁+𝐵𝐺3 (36)
and
𝐻𝑀𝐿 =𝑆𝑉+𝐵𝑉2 −𝑆𝐺+𝐵𝐺2 (37)
where SV is Small Value, SM is Small Neutral, SG is Small Growth, BV is Big Value, BN is Big Neutral, and BG is Big Growth portfolios.
Thus, the SMB is the difference in returns between a portfolio consisting of small stocks and a portfolio consisting of large stocks, whereas the HML is the
difference in returns between a portfolio with high book-to-market value stocks and a portfolio with low book-to-market growth stocks. The construction of the two last FF factors CMA and RMW is described in detail in the following subsection.